Prove that for all we have

Before proceeding with the proof, we recall the second mean-value theorem for integrals (Theorem 5.5 on p. 219 of Apostol). For a continuous function on the interval if has a continuous derivative which never changes sign on the interval then there exists a such that

* Proof. * Now, we want to apply the mean-value theorem above with and . Since is continuous everywhere , it is continuous on any interval . Then,

is continuous for all (so, in particular, for all ). Furthermore, since for all we have that for all . Thus, is continuous and never changes sign in any interval . Therefore, we can apply the mean-value theorem to conclude there exists a for any such that

But since for all we know for any and . Hence,